中文摘要：

为研究平面上以正六边形为基本集的周期函数的性质，定义了在平面上以矩形为基本集的半交错周期函数概念，并讨论了这2类函数之间的一些相互关系．由此可以揭示这2类函数的Fourier级数理论的一些联系．顺便给出一个关于Jackson不等式的精确结果．

英文摘要：

The authors study the properties of the regular hexagon periodic functions on the plane, and introduces the notion of rectangle self- alternative periodic functions on a plane. Moreover, the authors establish some interrelationship between two classes of functions, revealing connections between the Fourier series theory of these two function classes. In addition, an exact result of Jackson inequality was obtained.